# MomentAccumulator class

`MomentAccumulator`

accumulates the zeroth order moment, the first order normalized moments, and the
second order central normalized moments, in `N`

dimensions.

### Contents

Samples are added one by one, using the `Push`

method. Other members are used to retrieve the moments.

It is possible to accumulate samples in different objects (e.g. when processing with multiple threads),
and add the accumulators together using the `+`

operator.

## Constructors, destructors, assignment and conversion operators

- MomentAccumulator(dip::uint N)
- The constructor determines the dimensionality for the object.

## Functions

- void Reset()
- Reset the accumulator, leaving it as if newly allocated.
- void Push(dip::FloatArray const& pos, dip::dfloat weight)
- Add a sample to the accumulator.
`pos`

must have`N`

dimensions. - auto Sum() const -> dip::dfloat
- Sum of weights (zeroth order moment)
- auto FirstOrder() const -> dip::FloatArray
- First order moments, normalized
- auto SecondOrder() const -> dip::FloatArray
- Second order central moment tensor, normalized

## Operators

- auto operator+=(dip::MomentAccumulator const& b) -> dip::MomentAccumulator&
- Combine two accumulators

## Function documentation

### dip::FloatArray SecondOrder() const

Second order central moment tensor, normalized

The moments are stored in the same order as symmetric tensors are stored in an image (see dip::Tensor::Shape). That is, fist are the main diagonal elements, then the elements above the diagonal, column-wise. This translates to:

- 2D: xx, yy, xy
- 3D: xx, yy, zz, xy, xz, yz
- 4D: xx, yy, zz, tt, xy, xz, yz, xt, yt, zt
- etc.

The second order moment tensor is defined as:

where is the identity matrix ( ), is the weight of point , and is its position. In 2D, this leads to:

In 3D, it leads to:

The equations above represent the second order moments, we compute instead the central moments, and normalize them by the sum of weights.